One-Loop Quantum Gravity from the N = 4 Spinning Particle

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The main step forward in constructing consistent worldline descriptions of Yang-Mills [11] and gravity [1] was to realize that the corresponding BRST systems are consistent only upon a suitable truncation of the Hilbert space. In the case of the = 2 spinning particle this has an immediate field theoretic explanation, as p-forms admit Yang-MillsN interactions only for p = 1 . Similarly, the minimal constraint in the = 4 case is U(1) U(1) , that leaves the massless NS-NS spectrum of closed string theory, andN fits with the field theory× result of [12]. While this is not a problem for the BRST cohomology itself, nor for tree level amplitudes, one has to find the correct way to implement the projection at one-loop level, since in principle all the unwanted states pro! pagate in the loop. Moreover, the path integral on the circle is usually derived from gauge fixing a worldline action with local (super)symmetries, based on a first class constraint algebra. In the presence of a non-trivial gravitational background, the constraint algebra is obstructed and not first class anymore. A more appropriate way of thinking about the model is thus to consider it as a genuine BRST system from the start, regardless of being derived from a gauge invariant classical predecessor. Our main goal here will be to determine the measure on the moduli space implementing the correct projection on the pure gravity contribution, thereby defining the path integral on the circle. Thus, we study again the =4 spinning particle from this new prospective. First we consider the version defined by gaugingN the whole extended supersymmetry algebra of the worldline. In particular, we analyze the path integral on the loop (a worldline with the topology of a circle), constructed in [13], and dissect it to see how the gauge symmetries project the full Hilbert space to the one of the spin 2 particle, which remains as the only propagating degree of freedom. Studying the role played by the measure on the moduli space, left over by the gauge-fixing, allows us to modify the measure to achieve an improved projection. The latter has the virtue of working in any spacetime dimensions, allowing also more general couplings to curved backgrounds. This modification of the measure on moduli space is interpreted as due to the gauging a parabolic subgroup of the SO(4) R-symmetry group, supplemented by appropriate Chern-Simons couplings on the worldline. The model is eventually tested on curved backgrounds, were it is found to reproduce the correct results for the diverging part of the graviton one-loop effective action [14, 15].

2 Anatomy of degrees of freedom

We are going to review the quantization on the circle of the free =4 spinning particle with various gaugings of the SO(4) R-symmetry algebra. In particular,N we focus on the precise way the path integral extracts the physical degrees of freedom from the Hilbert space. This serves as a guiding principle for the quantization of the corresponding non-linear sigma model which couples a spin 2 particle (the graviton) to background gravity, and we use it to compute terms in the graviton one-loop eﬀective action in arbitrary dimension, checking that it gives the expected gauge invariant (BRST invariant) result in four dimensions.

2.1 Warm up: =2 and p-forms N Before analyzing the case of the =4 spinning particle, relevant for gravity, we shall review the quantization on the circle of the N =2 particle, describing differential forms [16, 17], in order to display how the gauging of worldlineN supersymmetries extracts the physical degrees of freedom in a covariant way, that is related to the BV treatment of gauge theories in target space. This is maybe not surprising, since the BRST wavefunction of the particle contains the full BV spacetime spectrum, and the first-quantized BRST operator serves as BRST-BV differential in target space [18–21]. The action for the free =2 spinning particle in phase space reads N

µ µ 2 S = dt pµx˙ + i ψ¯ ψ˙µ ep iχ ψ¯ p iχ¯ ψ p a(ψ ψ¯ q) , (2.1) − − · − · − · − Z

2 where the one dimensional supergravity multiplet (e, χ, χ,a¯ ) gauges worldline reparametrizations, supersymmetries and U(1) R-symmetry, respectively. Here “ ” means contraction over spacetime µ · µ µ indices µ, ν, ... while x (t) and pµ(t) are spacetime phase space variables, and (ψ (t), ψ¯ (t)) are worldline fermions, analogous to the RNS fermions of the spinning string. In order to project on p-form gauge fields we have included a Chern-Simons coupling1 q = p +1 D that converts the − 2 classical constraint (ψ ψ¯ q) into the operator (Nˆψ p 1) [17]. The contribution fro! m the ghosts will then shift the· eigenvalue− to p of the p-form− gauge− field. The euclidean action in configuration space becomes

1 µ µ µ 2 µ S = dτ (x ˙ χψ¯ χψ¯ ) + ψ¯ (∂τ + ia)ψµ + iqa , (2.2) 4e − − Z upon Wick-rotating the gauge ﬁeld a ia to maintain the U(1) group compact. On the circle we gauge ﬁx the supergravity multiplet→ as −(e, χ, χ,a¯ ) (T, 0, 0,θ) and the nontrivial ghost action is given by →

S = dτ β¯(∂τ + iθ)γ + β(∂τ iθ)¯γ , (2.3) gh − Z 1 for the system of bosonic superghosts. The partition function (that multiplied by 2 corresponds to the QFT eﬀective action) is then given by −

∞ dT 2π dθ Z = Z (T ) ,Z (T ) := Dx DψDψ¯ DβDγDβD¯ γe¯ −Sgf −Sgh , (2.4) p T p p 2π Z0 Z0 ZP ZA ZA

where Sgf denotes the gauge ﬁxed version of (2.2), and the subscripts on the functional integrals denote (anti)-periodic boundary conditions. In order to unveil the spacetime gauge structure, it is instructive to recast the density Zp(T ) in operator terms as

2π dθ −2 ˆ ˆ Z (T )= 2cos θ Tr e−T H eiθ(Nψ−p−1) , (2.5) p 2π 2 Z0 h i where Hˆ =p ˆ2 for the free theory, and the trace is over the Hilbert space consisting of differential forms of arbitrary degree contained in the ψ Taylor coefficients of wavefunctions ω(x, ψ) . The θ −2 factor of 2cos 2 comes from the path integral over the SUSY ghosts, and is responsible for the spacetime gauge structure, as we will briefly review. To proceed further, let us notice that the Hilbert space can be decomposed as a direct sum according to the form degree, i.e. the eigenvalues D of Nˆψ, as = n . Accordingly, the trace can be decomposed as well and we shall denote H n=0 H L −T Hˆ tn(T ) := Trn e . (2.6) h i D 1 For the free particle one has tn(T ) = n (4πT )D/2 , where the T -factor corresponds to the free particle position, while the binomial simply gives the corresponding number of independent components of an antisymmetric tensor of rank n . One can now use a Wilson line variable z := eiθ and find D p dz 1 n−p k Zp(T )= tn(T ) z = ( ) (k + 1) tp−k(T ) , (2.7) − 2πiz (z + 1)2 − γ n=0 k I X X=0 where we have slightly deformed the contour z = 1 to exclude the pole in z = 1 . The above result coincides with the decomposition one would| | obtain from the BV action in field− theory: the contribution for k =0 being the gauge p-form, k =1 the (p 1)-form ghost, k =2 the first ghost for ghost and so on. It is clear, from the way the expansion− is extracted from the above formula, z that (z+1)2 is the crucial factor for obtaining the correct residues and, as anticipated, it comes from the superghosts2. In order to treat the SUSY ghost sector in a similar footing to the ψ “matter”

1 ¯ 1 ¯ ¯ ˆ D We use a symmetric (Weyl) ordering of the quantum operators, e.g. ψ · ψ = 2 (ψ · ψ − ψ · ψ) → Nψ − 2 , where ˆ · ∂ − Nψ = ψ ∂ψ is the ψ number operator. This matches with the path integral regularization we use. 2This should be expected, since in the light-cone formulation the local supersymmetries explicitly remove the unphys- ical polarizations.

3 sector, we now rewrite the same density by undoing the path integral over the βγ-system:

2π dθ −T Hˆ iθ(N−ˆ p) Nˆγ +Nˆβ −T Hˆ Nˆγ +Nˆβ Zp(T )= TrBRST e e ( ) = TrBRST e δ( ˆ p)( ) , (2.8) 0 2π − N − − Z h i h i where the symbol δ( ˆ p) is a Kronecker delta, that selects the contribution of sectors with ˆ - N − µ N occupation number equals to p. Here ˆ := ψ ψ¯µ + γβ¯ βγ¯ and we observe that on the BRST N − vacuum annihilated by γ¯ and β¯ it also takes the form ˆ = Nψ + Nγ + Nβ; the trace is over the BRST Hilbert space3 with vacuum annihilated by β¯ andN γ¯ . We shall notice that the alternating signs appearing in (2.8) are due to the antiperiodic boundary conditions in a bosonic path integral, and have the spacetime interpretation of assigning negative contributions to the eﬀective action (and so to the degrees of freedom) to ﬁelds of odd ghost number. To exemplify the above, let us consider the case of p =2 : The BRST wavefunction at ˆ =2 is given by4 N µ ν µ ∗ µ 2 ∗ 2 Bµν ψ ψ + λµ ψ β + λµ ψ γ + φγβ + λβ + λ γ (2.9)

∗ where Bµν is the two-form gauge ﬁeld, (λµ, λµ) are the ghost-antighost vectors, φ is an auxiliary scalar and (λ, λ∗) is the ghost for ghost paired with its canonical conjugate, giving immediately

Z (T )= t (T ) 2t (T )+3t (T )= 1 (D−2)(D−3) , (2.10) 2 2 − 1 0 (4πT )D/2 2 that corresponds to the physical transverse degrees of freedom of a two-form.

2.2 =4 : gravitons and NS-NS spectrum N We turn now to the same analysis for the case of the =4 spinning particle, relevant for gravity, with various gaugings of the R-symmetry group SON(4) . The corresponding phase space action reads µ i µ 2 i i r S = dt pµx˙ + i ψ¯ ψ˙i µ ep iχi ψ¯ p iχ¯ ψi p a Jr , (2.11) − − · − · − Z where i = 1, 2 is a U(2) index and Jr denotes the subset of SO(4) generators being gauged—we keep manifest only the symmetry U(2) SO(4). The states in the Hilbert space correspond to bi-forms, interpreted as gauge ﬁelds: ⊂

D µ1 µm ν1 νn ω(x, ψi)= ω (x) ψ ...ψ ψ ...ψ m n . (2.12) µ[m]|ν[n] 1 1 2 2 ∼  ⊗ m,n=0 m,n ( X M  Being interested in gravity we will always project the above spectrum to the subspace m = n =1 for the graviton to be identified with the symmetric and traceless component of ωµ|ν . In this respect, the most economic choice is to gauge the U(1) U(1) subgroup generated by × î Nî := ψî ψ¯ , index i not summed , (2.13) · where we used hats to stress that the above expression refers to operators in that given order. In D this case one can add two independent Chern-Simons couplings qi , that we choose as qi = 2 2 in order to project on the gravity sector. The spectrum consists of a graviton, a two-form and− a scalar, coinciding with the massless NS-NS sector of closed strings. On the circle, the two gauge fields ai give rise to the angular moduli (θ, φ) and, similarly to the =2 case, we obtain N 2π 2π dθ dφ −2 −2 ˆ ˆ ˆ Z (T )= 2cos θ 2cos φ Tr e−T H eiθ(N1−2)+iφ(N2−2) . (2.14) U(1)×U(1) 2π 2π 2 2 Z0 Z0 h i 3This coincides with the treatment given in [11] for spin one. 4Prior to integrating the (b, c) reparametrization ghosts, the BRST Hilbert space is doubled by the presence of c in the wavefunction. This corresponds to the full BV spectrum in spacetime, with all the antifields included. Here, having already integrated out the (b, c) system, the wavefunction corresponds to taking Siegel’s gauge b Ψ = 0 .

4 The Hilbert space, and so the trace, has an obvious double grading according to the eigenvalues of Nˆi . We will thus deﬁne −T Hˆ tm,n(T ) := Trm,n e , (2.15)

allowing to write h i

D dz dw z w n−2 m−2 ZU(1)×U(1)(T )= tn,m(T ) z w , (2.16) − 2πiz − 2πiw (z + 1)2 (w + 1)2 γ γ n,m=0 I I X where the Wilson line variables are given by z := eiθ and w := eiφ . The contour integrals are easily performed, yielding5 Z = t , 2 t , 2 t , +4 t , . (2.17) U(1)×U(1) 1 1 − 1 0 − 0 1 0 0 1 D D In the free theory one has tn,m(T )= tm,n(T )= (4πT )D/2 n m , thus giving the number of degrees of freedom Dof = (D 2)2 that corresponds to the transverse polarizations of the tensor U(1)×U(1) − ωµ|ν . The above partition function can be decomposed into its irreducible spacetime components:

Z = t , 4 t , +4 t , = [t , t , 2 t , ]+[t , 2 t , +3 t , ]+t , = Zh+ZB+Zφ (2.18) U(1)×U(1) 1 1− 1 0 0 0 1 1− 2 0− 1 0 2 0− 1 0 0 0 0 0 namely the spin two graviton, the two-form and a scalar. The corresponding degrees of freedom decompose accordingly as

D(D 3) (D 2)(D 3) Dof = (D 2)2 = − + − − +1 (2.19) U(1)×U(1) − 2 2

and coincide with the transverse polarizations of the (traceless) graviton hij , two-form Bij and scalar φ . In the following we will be interested in gauging larger subgroups of SO(4) in order to project out the two-form and/or the scalar field from the spectrum. In particular we will now analyze how the maximal gauging of the entire R-symmetry group, that was studied in [13] for general , extracts the graviton degrees of freedom. When the entire SO(4) group is gauged there is no roomN for Chern-Simons couplings. This fixes the Young diagram of the spacetime gauge field D−2 to have 2 rows, thus yielding a graviton only in D = 4 . Restricting to four dimensions, the partition function is given by (see [13] for the derivation)

2π 2π 1 dθ dφ −2 −2 2 2 Z (T )= 2cos θ 2cos φ 2i sin θ+φ 2i sin θ−φ SO(4) 4 2π 2π 2 2 2 2 Z0 Z0 (2.20) ˆ ˆ ˆ Tr e−T H eiθ(N1−2)+iφ(N2−2) . × h i In the above expression the cosine factors, as in the previous case, come from the path integral over the SUSY ghosts. The sine factors result instead from the path integral of the non-abelian ghosts of SO(4) . In particular, the factor containing the diﬀerence of the angles corresponds to the U(2) j ¯j subgroup generated by Ji := ψi ψ , while the other factor, depending on the sum of the angles, is ij · i j related to the gauging of K¯ := ψ¯ ψ¯ (trace operator) and Kij := ψi ψj (insertion of the metric). In terms of Wilson line variables we· have ·

D 1 dz dw z w n−2 m−2 ZSO(4)(T )= p(z, w) tn,m(T ) z w , (2.21) 4 − 2πiz − 2πiw (z + 1)2 (w + 1)2 γ γ n,m=0 I I X where the function (zw 1)2(z w)2 z w 2 1 1 p(z, w)= − − =4 2zw 2 + + z2 + w2 + + (2.22) z2w2 − − w z − zw z2 w2 contains all the dependence on the non-abelian gauging. We shall now compute the contributions to (2.21) of the various components of p(z, w) in order to see how the projection on the graviton is

5For brevity we omit the dependence on T .

5 achieved

4 t , 4 t , +4 t , Z → 1 1 − 1 0 0 0 ≡ U(1)×U(1) 1 1 2zw t , Zφ − → − 2 0 0 ≡− 2 z w 2 + (t , 2 t , +3 t , ) ZB − w z → − 2 0 − 1 0 0 0 ≡− 2 1 1 (t , +4 t , +9 t , 4 t , +6 t , 12 t , ) ZA −zw → − 2 2 2 1 1 0 0 − 2 1 2 0 − 1 0 ≡− 2 2,2 z2 + w2 0 →

1 1 1 1 1 + (t , 2 t , +3 t , 4 t , ) (t , 2 t , +3 t , 4 t , ) ZA , z2 w2 → 2 3 1 − 2 1 1 1 − 0 1 − 2 3 0 − 2 0 1 0 − 0 0 ≡ 2 3,1 (2.23) where the arrows mean that the given monomial yields the corresponding contribution upon inte- gration over the moduli z and w. The first three contributions are clear from what we have discussed so far. One can see, for instance, that the third contribution removes the two-form from the reducible partition function ZU(1)×U(1) , while the second contribution removes only “half” of the scalar field. To explain the interpretation of the second half of (2.23), let us introduce (p, q) gauge bi-forms. By a gauge bi-form Ap,q we denote the tensor field

µ1 µp ν1 νq Ap,q(x, ψi)= Aµ[p]|ν[q](x) ψ1 ...ψ1 ψ2 ...ψ2 p q , (2.24) ∼  ⊗ (  with gauge symmetries  µ δAp,q = d1λp−1,q + d2ξp,q−1 , di := ψi ∂µ (2.25) and gauge invariant curvature Fp+1,q+1 = d1d2Ap,q . The corresponding partition function can be obtained from the previous case of U(1) U(1) gauging by modifying the Chern-Simons couplings to q = p +1 D and q = q +1 D , yielding× 1 − 2 2 − 2 p,q k+l ZA = ( ) (k + 1)(l + 1)tp k,q l , (2.26) p,q − − − k,l X=0 that justifies the identifications made in (2.23). Let us now consider the contribution of ZA2,2 to ZSO(4). By performing a double Hodge dualization of the field A2,2 we can see that it is dual to a scalar (recalling that we are in four dimensions):

curvature Hodge dual potential A , F , F , A , φ . (2.27) 2 2 −−−−−−→ 3 3 −−−−−−−→ 1 1 −−−−−→ 0 0 ≡ Similarly, one can see that A3,1 is a non-propagatinge ﬁeld e e

curvature Hodge dual potential A , F , F , (2.28) 3 1 −−−−−−→ 4 2 −−−−−−−→ 0 2 −−−−−→ ∅ with zero degrees of freedom. We can now put togethere the results of (2.23) and, using the decomposition (2.18), obtain

1 1 Z = Zh + (Zφ Ze)+ ZA . (2.29) SO(4) 2 − φ 2 3,1 We can thus conclude that the effective action generated by the full SO(4) gauging corresponds to the graviton plus topological terms. The latter vanish in the free theory, corresponding to zero degrees of freedom, but do contribute to the effective action on non-trivial backgrounds. Our goal is to find a modified one-loop measure for the path integral that projects exactly onto the graviton z w state. In the measure (2.21) the factors of (z+1)2 (w+1)2 and p(z, w) play very different roles: as we

6 have previously mentioned, the former corresponds to the gauging of worldline supersymmetries, and is responsible of the spacetime gauge symmetry that ensures unitarity. The latter factor, related to the gauging of the R-symmetries, performs algebraic projections on the spectrum and is the only one that we will modify. To begin with, we notice that the SO(4) projector (2.22) has the manifest symmetry p(z, w)= p(1/z, 1/w) and can be written as z w p(z, w)= P (z, w)+ P (1/z, 1/w) , P (z, w)=2 + 2zw + z2 + w2 . (2.30) − w z − By looking at the decomposition (2.23), it is clear that the unwanted contributions from dual ﬁelds come from the term P (1/z, 1/w) , that indeed entails double Hodge dualization. We will thus propose to use 2P (z, w) instead of p(z, w) as an ansatz for the graviton projector:

1 dz dw z w −T Hˆ Nˆ1−2 Nˆ2−2 Zgrav(T )= 2 2 P (z, w)Tr e z w . (2.31) 2 γ− 2πiz γ− 2πiw (z + 1) (w + 1) I I h i This ansatz for the projector can be related to a diﬀerent gauging of the R-symmetry algebra. First, notice that P (z, w) can be written in factorized form as

(zw 1)(z w)2 2 P (z, w)= − − , or P (θ, φ)=2i sin θ+φ 2i sin θ−φ exp i θ+φ . (2.32) zw 2 2 2 We now consider the case of gauging the parabolic subalgebra of so(4) consisting of the u(2) j ¯ ij subalgebra, generated by Ji , plus the trace K . We are thus excluding, compared to the maximal gauging, the insertion of the metric Kij , which causes in the measure the depletion of one sine factor, yielding

2 θ+φ θ−φ Pparab(θ, φ)=2i sin 2 2i sin 2 , (2.33) P (θ, φ) = exp i θ+φ P (θ, φ) . (2.34) ⇒ 2 parab The extra exponential factor can be accounted for, since the parabolic gauging allows for a Chern- Simons term proportional to the diagonal U(1) :

gauge fix S = iq dτ ai iq(θ + φ) . (2.35) CS i −→ Z By an appropriate choice of q it is possible to reproduce the graviton projector P (θ, φ) and also to go in arbitrary dimensions. In fact, recall that the shift in the number operators in (2.21) is in D general Nî . By choosing the Chern-Simons coupling as − 2 D 3 D 1 q = − =2 , − 2 − 2 − 2 1 ˆ D ˆ the 2 part produces P (θ, φ) from Pparab(θ, φ) , while the rest modifies the shifts Ni 2 to Ni 2 in arbitrary dimensions. − −

3 One loop gravity eﬀective action

In the present section we apply the method described above to represent the one-loop eﬀective action for pure Einstein-Hilbert gravity, and compute it in a local expansion up to quadratic orders in the curvature6. The starting point is the SO(4)-extended locally supersymmetric spinning particle, whose phase-space action corresponds to the curved deformation of action (2.11). It was previously studied in [22], where a suitable BRST gauge ﬁxing, involving the whole R-symmetry group, was analyzed.

6For massless particles the local expansion of the one-loop eﬀective action is not permitted, except for identifying the divergences which indeed are local. Here we consider precisely those terms.

7 In the present approach, we instead consider the gauge ﬁxing of the parabolic subgroup discussed above, which corresponds to the phase space action

µ i µ i i 1 ij j i i S[z, E; g]= dt pµx˙ + i ψ¯ ψ˙i µ eH iχi ψ¯ π iχ¯ ψi π aij K¯ ai (J iqδ ) , − − · − · − 2 − j − j Z h (3.1)i

i j where z = (x, p, ψ, ψ¯), E = (e,χi, χ¯ ,aij ,ai ), and

a ib πµ = pµ ωµab ψ ψ¯ (3.2) − i µν a ib c jd H = g πµπν Rabcd ψ ψ¯ ψ ψ¯ . (3.3) − i j The one-loop effective action thus corresponds to the circle path integral of the previous action. i The antiperiodic gravitini χi, χ¯ are again gauge-fixed to zero, whereas the einbein gets gauge- fixed to its modulus, T , which is interpreted as the Schwinger proper-time, and the gauge fields of the parabolic subgroup are fixed to the two angles of the associated Cartan torus, with the Faddeev-Popov determinant given by the expression (2.33). In euclidean configuration space we thus have 1 ∞ dT Γ[g]= Z(T ) , (3.4) −2 T Z0 with 1 2π dθ 2π dφ θ −2 φ −2 Z(T )= 2cos 2cos P (θ, φ) e−iq(θ+φ) 2 2π 2π 2 2 parab Z0 Z0 1 1 µ ν ia j j (3.5) x DψDψ¯ exp dτ gµν x˙ x˙ + ψ¯ (Dτ δ Λi )ψja × D − 4T i − ZP ZA Z0 n h a b c d T Rabcd ψ ψ¯ ψ ψ¯ +2T V , − · · im io where Λ j = θ 0 , D is the covariant derivative in the multispinor representation of the Lorentz i 0 φ τ group, and the scalar potential V = D+2 R is an order ¯h2 improvement term, generated at im − 8(D−1) the quantum level from the anticommutator of the supersymmetry generators. The constraints are not first class, but the expression (3.4) coincides with the one coming from the BRST system of ref. [1], considered as the starting point for the path integral. The BRST system, and thus the path integral, is only consistent (upon projection) on Einstein manifolds: Rµν = λgµν , which is the class of allowed backgrounds [1]. The above curved space particle path integral involves a nonlinear sigma model action, whose perturbative (short time) evaluation needs a careful regularization. This is well studied (see ref. [23] for a review), even in models with extended supersymmetries [24]. In particular, the use of worldline dimensional regularization implies the inclusion of a counterterm potential which, for = 4, is 1 N VCT = 8 R and, together with the improvement term gives 1 D +2 3 V = R = R =: ωR . (3.6) 8 − 8(D 1) −8(D 1) − − With these prescriptions, and with the Feynman rules described in [22], we obtain

1 2π dθ 2π dφ θ D−2 φ D−2 Z(T )= 2cos 2cos P (θ, φ) e−iq(θ+φ) 2 2π 2π 2 2 parab Z0 Z0 g D √ −Sint d x D e × (4πT ) 2 Z D E g D √ −Sint =: d x D e , (3.7) (4πT ) 2 Z DD EE where

e−Sint =1 T 5 +2ω + 1 cos−2 θ + cos−2 φ R − − 12 4 2 2 D E 8 2 + T 2 1 5 +2ω + 1 cos−2 θ + cos−2 φ R2 2 − 12 4 2 2 n + 1 1 cos−4 θ + cos−4 φ + 1 cos−2 θ + cos−2 φ R2 − 180 − 8 2 2 8 2 2 ab 2 + 1 1 cos −4 θ + cos−4 φ 1 cos−2 θ + cos−2 φ + 1 cos−2 θ + cos−2 φ R2 180 − 32 2 2 − 48 2 2 16 2 2 abcd 9 +ω + 1 cos−2 θ + cos −2 φ 2R + O(T 3) , (3.8) − − 120 3 24 2 2 ∇ o and ... is the modular integral of the path integral average. As above we ﬁnd it convenient to rewritehh theii modular integrals in terms of complex variables rather than angles. We thus have 1 dz dw (1 + z)D−2 (1 + w)D−2 e−Sint = (zw 1)(z w)2 2 2πiz 2πiw z2 w2 − − DD EE I I 5 z w 1 T 12 +2ω + (1+z)2 + (1+w)2 R ( − − 2 2 1 5 z w 2 + T +2ω + 2 + 2 R 2 − 12 (1+z) (1+w) h 1 z2 w2 1 z w 2 + 2 4 + 4 + 2 + 2 R − 180 − (1+z) (1+w) 2 (1+z) (1+w) ab 1 1 z2 w2 1 z w z w 2 2 + 4 + 4 2 + 2 + 2 + 2 R 180 − 2 (1+z) (1+w) − 12 (1+z) (1+w) (1+z) (1+w) abcd 9 ω 1 z w 2 3 120 + 3 + 6 (1+z)2 + (1+w)2 R + O(T ) , (3.9) − − ∇ ) i which, after performing the contour integrals, yields

D(D 3) 5D2 35D + 12 e−Sint = − 1+ T R − 2 ( 12(D 1)(D 3) DD EE − − 25D4 275D3 + 232D2 432D + 288 D2 183D 720 + T 2 R2 − − R2 − − 288D(D 1)2(D 3) − ab 180D(D 3) " − − − D2 33D + 540 9D2 61D + 22 + R2 − + 2R − + O(T 3) . (3.10) abcd 180D(D 3) ∇ 120(D 1)(D 3) − − − # ) For D =4 it reduces to

−Sint 8 2 31 2 359 2 53 2 13 2 3 e = 2 T R + T R + Rab + Rabcd R + O(T ) , (3.11) ( − 3 "−18 90 45 − 30∇ # ) DD EE that on Einstein manifolds, coincides with the expression found in ref. [10], namely

−Sint 8 2 29 2 53 2 3 e = 2 T R + T R + Rabcd + O(T ) , (3.12) ( − 3 "−40 45 # ) DD EE 2 where Rabcd can be identified with the four-dimensional Euler density of Einstein manifolds. The different powers of T , once inserted into (3.7) and (3.4), give rise to the quartic, quadratic and logarithmic divergences of the graviton one-loop effective action. In spacetime dimensional regularization the first two terms are invisible, and the logarithmic divergence vanishes for null cosmological constant, thus reproducing the famous result by ’t Hooft and Veltman [14]. With a cosmological constant, the logarithmic divergence coincides with the one found in [15]. More generally, all these divergences reproduce those computed in [10]. The on-shell expression given in eq. (3.12) is gauge independent, and it is a benchmark for any correct calculation in perturbative quantum gravity. Thus, for completeness, we also report the same result for a graviton on a D-dimensional Einstein manifold D(D 3) D(5D2 35D + 12) e−Sint = − + T R − 2 24(D 1) DD EE − 9 125D5 1383D4 + 2640D3 + 664D2 8616D + 5760 + T 2 R2 − − 2880D(D 1)2 " − 2 2 D 33D + 540 3 + Rabcd − + O(T ) , (3.13) 360 # where the first term gives the number of its degrees of freedom.

4 Conclusions

We have constructed a modified =4 spinning particle that is able to describe the pure graviton on Einstein spaces. It is identifiedN by gauging only a parabolic subgroup of the SO(4) R-symmetry group of the particle, and adding suitable Chern-Simons couplings on the worldline. The gauging of parabolic subgroups have already been used in worldline models for higher spin particles, as in ref. [25]. They give rise to worldline actions that are not real, but which do not seem to produce any pathology in the spacetime intepretation of the theory, in that the sole purpose of the R-symmetry gauging is to perform an algebraic projection on the spacetime spectrum to achieve different degrees of reducibility. Similarly, the use of Chern-Simons couplings on the worldline helps to insert projectors in the Hilbert space, so to leave only a desired subsector containing the physical states. They appeared, for example, in the worldline description of differential forms [17, 26, 27]. Our model has been able to reproduce the correct divergences of one-loop quantum gravity. Together with the BRST construction of ref. [1], few working tools are now available to address quantum gravity from a worldline perspective. We stress again that the model developed in this paper can be naively obtained by gauging the worldline supersymmetries, together with appropriate subalgebras of the R-symmetry. However, the classical supersymmetry algebra is broken by the background curvature, and the action has to be seen as a quantum BRST model, whose consistency is guaranteed by the truncation of the Hilbert space. On the reduced Hilbert space the BRST charge is nilpotent only when the background is on- shell according to Einstein’s equation [1], showing that the “prediction” of target space equations from quantum consistency of the first-quantized theory is not a peculiarity of string theory, as already confirmed by the earlier work of ref. [11]. It would be useful to extend further these constructions and find more applications of worldline methods to quantum gravity, for instance extending the present description to non-commutative spaces [28]. An immediate generalization of the present model consists in weakening the R- symmetry constraint to U(1) U(1) , thereby letting all the massless NS-NS particles circulate in the loop. Further coupling× the model to a background B-field and dilaton [29] would allow to obtain the one-loop effective action for the whole NS-NS massless sector of string theory. An alternative to the present formulation in terms of fermionic oscillators is to use the Sp(2) particle, with bosonic oscillators. The advantage, once a consistent BRST system is found, would be to easily treat all spins at once, by just modifying a suitable Chern-Simons coupling. This should also reproduce the well known no-go results for minimal coupling higher spin fields to gravity [30,31].

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